$\dfrac{dy}{dx}=-\dfrac{x}{y}$ Is $y=\sqrt{10-x}$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answer: In order to find whether $y=\sqrt{10-x}$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $y$, we need to find the corresponding $\dfrac{dy}{dx}$ expression to substitute into the equation: $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{d}{dx}\left[\sqrt{10-x}\right] \\\\ &=-\dfrac{1}{2\sqrt{10-x}} \end{aligned}$ Now we substitute ${y=\sqrt{10-x}}$ and ${\dfrac{dy}{dx}=-\dfrac{1}{2\sqrt{10-x}}}$ into the equation: $\begin{aligned} {\dfrac{dy}{dx}}&=-\dfrac{x}{{y}} \\\\ {-\dfrac{1}{2\sqrt{10-x}}}&\stackrel{?}{=}-\dfrac{x}{{\sqrt{10-x}}} \\\\ -\dfrac12&\neq -x \end{aligned}$ We did not obtain equivalent expressions on each side. In conclusion, no, $y=\sqrt{10-x}$ is not a solution to the differential equation.